Q. J. R. Meteorol. Soc. (2004), 130, pp. 2087–2103
doi: 10.1256/qj.03.161
Flux–gradient relationship, self-correlation and intermittency in
the stable boundary layer
By CHERYL L. KLIPP and LARRY MAHRT∗
Oregon State University, Corvallis, USA
(Received 21 August 2003; revised 3 March 2004)
S UMMARY
The correlation between dimensionless shear φm and dimensionless height z/L, where L is the Obukhov
length, for stable conditions is strongly influenced by self-correlation for the present datasets. This effect is quite
large for stronger stability but still significant for near-neutral conditions. A conditional analysis of nocturnal
stable boundary-layer data, where ‘non-turbulent’ parts of the record are removed, reduces the impact of nonstationarity and therefore reduces the scatter. The conditional analysis also reduces the relative importance of
self-correlation. Difficulties with estimating self-correlation are also discussed.
K EYWORDS: CASES99 Monin–Obukhov similarity Nocturnal boundary layer z-less similarity
1.
I NTRODUCTION
Nocturnal surface cooling induces stratification and partial suppression of turbulence. Values of boundary-layer depth range from a few hundred metres thick with
strong winds (large shear generation) and cloudy skies, to only a few metres thick
with weak winds and clear skies. During stable conditions, intermittent turbulence often
develops, where the strength of the turbulence varies between moderate and very weak,
typically on time-scales of a few minutes to a few tens of minutes. Some attribute the
intermittency to external forcings such as gravity waves (Finnigan and Einaudi 1981),
density currents (Soler et al. 2002), low-level jets (Banta et al. 2002), or other mesoscale
phenomena (Sun et al. 2002), others to interactions between the turbulence and local
mean gradients (Derbyshire 1999). Intermittency is seldom clearly defined and therefore
is sensitive to detection criteria.
For strongly stable conditions, where intermittency is common, the flux–gradient
relationship shows considerable scatter compared to more weakly stable conditions.
Published formulae for the flux–gradient relationship are generally not intended for
strongly stable cases, but are none the less used by numerical models for all stabilities.
In addition, the occurrence of the surface friction velocity in both the Obukhov
length and the non-dimensional shear may lead to considerable self-correlation.
Such self-correlation produces correlation of the same sign as the physically expected
correlation for stable conditions and can therefore lead to false confidence in
Monin–Obukhov similarity theory. Self-correlation due to a common divisor has been
recognized by the statistics community since Karl Pearson first promoted linear correlation as a scientific tool in the late nineteenth century (Aldrich 1995). It has been noted
as a problem in the atmospheric boundary-layer community since at least the late 1970s
(Hicks 1978a,b) and subsequently pursued by Hicks (1981), Andreas (2002), Mahrt
et al. (2003) and others. Self-correlation is usually referred to as spurious correlation
in the statistics literature. This study uses multiple datasets to examine the role of selfcorrelation and reconsiders the method for such evaluation.
∗ Corresponding author: College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis,
OR 97331, USA. e-mail: mahrt@coas.oregonstate.edu
c Royal Meteorological Society, 2004.
2087
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C. L. KLIPP and L. MAHRT
2.
M ONIN –O BUKHOV SIMILARITY THEORY
The flux–gradient relationship in the surface layer is usually posed in terms of
Monin–Obukhov similarity theory . This similarity theory can be derived from the Buckingham Pi groups (Stull 1988; Barenblatt 1996) obtained using ∂U /∂z, u∗ , (w θ )0 ,
g/θ , and z as the fundamental variables. Here the wind vector is (u, v, w), U is the
mean wind, z is the instrument elevation, primes indicate deviations from the mean,
u∗ = {(u w )20 + (v w )20 }1/4 is the friction velocity, (w θ )0 is the surface heat flux, g is
the acceleration due to gravity, and θ is the mean potential temperature. The results are
a dimensionless height z/L, where the Obukhov length L is
L=−
θ
u3∗
,
κg (w θ )0
and a dimensionless shear
κz
φm =
u∗
∂U
∂z
(1)
,
(2)
where κ is the von Karman constant.
One test of Monin–Obukhov similarity theory compliance is whether or not the
scaled variables are universal functions and have no additional dependencies on other
variables, such as those surveyed by Mahrt et al. (2003). Previous observations indicate
the non-dimensional shear for stable conditions to be approximately φm = 1 + β(z/L)
for z/L > 0. Published values for β usually fall in the range of 2.5–5.0 and are usually
restricted to z/L values less than 2 (e.g. Högström 1988; Sorbjan 1989; Högström
1996; Howell and Sun 1999; Vickers and Mahrt 1999). Holtslag and De Bruin (1988)
and Beljaars and Holtslag (1991) propose a nonlinear model for large values of z/L.
The relationship between φm and z/L shows considerable scatter, especially for large
positive values of z/L.
The Monin–Obukhov relationship (Eqs. (1) and (2)) becomes local scaling if
the heat flux and stress at level z are significantly different from the surface values
(Nieuwstadt 1984). When the instrument at level z is in the surface layer, Monin–
Obukhov surface-layer scaling and local scaling are approximately the same. If z is
above the surface layer, then the fluxes are significantly smaller than surface fluxes
and Monin–Obukhov similarity does not apply. Much of the data in this study fall into
this category, in which case, Monin–Obukhov similarity theory does not apply and the
relationships default to local similarity theory.
3.
DATA
We analyse data from the main 60 m tower in CASES99 (Poulos et al. 2002; Sun
et al. 2002) in October of 1999 for the nocturnal period 2000–0500 CST (UTC−6 h)
and exclude hours when the mean wind is through the tower (205◦ –335◦). The 20 Hz
data from sonic anemometers, deployed at 10, 20, 30, 40, 50 and 55 m, are used for
this analysis. Additional sonic anemometers at 1.5 and 5.0 m were deployed on a small
tower placed 10 m away. Each sonic anemometer recorded three-dimensional winds
and sonic virtual temperature. The sonic anemometer at 1.5 m was moved to 0.5 m on
19 October. Propeller anemometers recorded 1 Hz wind data at 15, 25, 35, and 45 m.
Conditions range from near-neutral with relatively strong turbulence to very stable with
weak turbulence and/or intermittent turbulence.
FLUX–GRADIENT RELATIONSHIP IN THE SBL
2089
(a) Flux calculations
In calculating turbulent fluxes, we must choose an appropriate time-scale over
which to compute the means and resulting deviations from the means. If too short a timescale is chosen, the flux will be underestimated; if too long, the flux will be contaminated
by mesoscale fluctuations and therefore large random flux errors. The ideal averaging
window for flux calculation depends on the stability and height above ground (Vickers
and Mahrt 2003).
Multi-resolution analysis of the CASES99 nocturnal data indicates that 100 s is
an acceptable averaging window except for weakly stable conditions (small z/L) where
longer windows might be more suitable. The φm and z/L values for these small z/L data
points change by less than 5% when using 100 s windows instead of longer windows
(5 min or more) owing to some cancellation of errors associated with underestimation
of both w θ and u∗ . The net error is generally less than the estimated random flux
error. For simplicity, a short 100 s window is used throughout this paper to calculate
the fluctuations because they provide optimal flux calculations for the more stable
conditions (large z/L), the focus of this paper, while having small effect on the weakly
stable data. Fluxes are then averaged over one hour to reduce random flux errors.
(b) Gradients
Estimated vertical gradients of mean wind and temperature profiles are sensitive
to the calculation method, especially near the surface. We compare the performance of
several methods and find a local fitting technique preferable to more traditional methods
of gradient estimation. The mean wind is expected to follow a log-linear profile only
in the surface layer. The surface layer, normally considered to be about the lowest
10% of the boundary layer, may be only a few metres deep, or less, for many of the
CASES99 profiles (Mahrt and Vickers 2003). A log-linear fit over the entire tower height
is inappropriate and does not provide accurate gradient estimates when the surface
layer is significantly shallower than the tower height. A log-linear fit over the entire
tower height is only slightly improved by removing the surface constraint U = 0 at
z = z0 = 3 cm (Fig. 1(a)). This value for the roughness length was calculated using only
data from near-neutral conditions.
A local log-linear fit better approximates the observed profiles (Fig. 1(b)) even
though such a function is not theoretically motivated above the surface layer. With the
local-fit method, the lowest profile fit uses the mean wind data from levels 1.5 (0.5), 5
and 10 m as well as the surface constraint U (z0 ) = 0. The gradient is evaluated only for
the middle levels, 1.5 (0.5) and 5 m. Next, the profile fit is applied to the mean winds
from levels 1.5 (0.5), 5, 10, and 20 m and the gradient evaluated for levels 5 and 10 m.
The process is sequentially repeated, stepping up the tower one instrument level at a
time. A separate fit is made with data from the four propeller anemometers at 15, 25,
35, and 45 m. For this profile fit, the gradient is evaluated for 20, 30 and 40 m.
Gradients calculated as finite difference and also as log-finite difference were found
to produce values comparable to the local log-linear fits at instrument levels of 10 m and
higher, but not at the lower levels. For these lower levels, the relationship between φm
and z/L using local-fit gradients yields φm (z/L = 0) = 1.0. For the same relationship
using finite-difference gradients, φm values can be significantly larger or smaller than
1.0 depending on the instrument level. Log-finite differencing yields values of φm closer
to 1.0 at z/L = 0, but still show a bias larger or smaller than 1.0 depending on the height
of the lower instrument levels. For this paper, all mean shears and temperature gradients
are evaluated with the local log-linear fit method.
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C. L. KLIPP and L. MAHRT
Figure 1. Vertical mean wind profile for 2200–2300 CST (UTC −6 h) 17 October 1999. Circles are sonic
anemometer data and squares are propeller data. (a) Traditional log-linear fit, with (—) and without (- -) surface
constraint. (b) Local fits to the same data.
TABLE 1.
T HE LINEAR - CORRELATION COEFFICIENT R BETWEEN φm AND z/L ( SEE TEXT )
Level:
0.5 m
1.5 m
5m
10 m
20 m
30 m
40 m
50 m
All φm , z/L > 0
No weak turbulence
0.58
0.75
0.90
0.73
0.52
0.83
0.69
0.85
0.79
0.71
0.64
0.95
0.70
0.92
0.35
0.89
4.
I NFLUENCE OF ELIMINATING VERY WEAK TURBULENCE
Much of the scatter inherent in nocturnal data can be eliminated by omitting hours
with weak turbulence and near-zero fluxes or local gradients (Table 1). Although gradients calculated from curves fitted to the wind data are potentially much more accurate
than gradients calculated as differences, values of ∂U /∂z < 0.001 s−1 are indistinguishable from zero. Small flux values are usually characterized by large relative random flux
errors (RFE), and may be dominated by mesoscale trend within the averaging window.
Using thresholds of standard deviation σw = 0.05 m s−1 , −w θ = 0.001 m K s−1 , and
∂U /∂z = 0.001 s−1 eliminates all data points with a relative RFE greater than 40%
and most data points with relative RFE > 15%. The RFE is calculated as the standard
deviation of the thirty-six 100 s fluxes used
√ to calculate the one-hour flux, divided by the
square root of 35, that is RFE = σflux / N − 1, where N is the number of data points.
The relative RFE is the RFE divided by the magnitude of the mean flux for that hour.
Application of these thresholds increases the linear-correlation coefficient R between φm and z/L. However, the linear-correlation coefficient is only a rough relative
measure of the scatter in this case since the number of points used to calculate R differs
between the set with all of the data and the set with no weak turbulence data. For the
1.5 and 20 m data, the outliers in the full dataset lie far from the main data cluster and
FLUX–GRADIENT RELATIONSHIP IN THE SBL
2091
therefore strongly influence the slope of the best-fit line and increase R with respect to
the less-scattered no-weak-turbulence data.
Selection of the threshold values for elimination of hours with very weak turbulence
is somewhat arbitrary. Setting the limit higher only slightly decreases scatter, but
eliminates a large number of data points. Setting the limit lower increases scatter.
Since hours with weak turbulence yield highly scattered data, often with large random
flux errors for individual records, the following analyses will only use data which satisfy
the above thresholds. Additional remaining scatter in extremely stable data may be due
to factors not considered here, such as systematic errors in the turbulent fluxes due to
flux loss from path length averaging.
5.
S ENSITIVITY OF LINEAR - CORRELATION COEFFICIENT TO SELF - CORRELATION
Both φm and z/L contain u∗ (Eqs. (1) and (2)), causing spurious self-correlation
between φm and z/L. Normally a very small value for the linear-correlation coefficient is considered to indicate no relationship between the variables being compared.
When these variables share a common divisor, however, even random data can produce
significant non-zero correlation (Kim 1999; Andreas 2002). This ‘random’ correlation
coefficient is a better reference point of no physical relation than R = 0.
The degree of self-correlation for data with a common divisor is related to the
range and distribution of the values of the fundamental variables. The range and
distribution are characterized by the coefficient of variation VX = σX /X (Kim 1999).
Larger coefficients of variation for the common divisor with respect to the coefficients
of variation for the other variables lead to larger self-correlation. In this application,
σX and X are the statistics for the whole range of data points collected and should not
be confused with the standard deviations used to calculate the uncertainty of individual
measurements.
Owing to the dependence of the magnitude of self-correlation on the distribution
of the data, it is important that synthesized random data have the same statistical distribution in order to make good comparisons with the actual data. By using randomized
actual data to evaluate self-correlation, rather than data synthesized by a random number
generator, the range and distribution of the original data are closely reproduced in the
random datasets (Kim 1999; Hicks 1978b, 1981).
(a) Random sampling with replacement
To test the possibility of significant self-correlation, we construct random datasets
by using the original observations as a pool of values to draw from at random. For a
given z, the fundamental variables that compose φm and z/L are w θ , u∗ and ∂U /∂z.
To construct the first random data point for level z, a heat-flux value is chosen at random
from the heat-flux data for that level, then a u∗ value is chosen at random from the
u∗ data, and a shear value is chosen at random from the shear data. The process is
repeated until the random dataset has as many points as the original data (Fig. 2).
Using the random data, new values for φm and z/L are computed and the linearcorrelation coefficient between them is computed (Mahrt et al. 2003). This process is
repeated 1000 times, that is, 1000 different random sets of data are constructed and
1000 correlation coefficients are calculated. Since the random data no longer retain
any physical connections between the fundamental variables, the average correlation
for these 1000 trials of random data, Rrand, is a measure of self-correlation due to
the common divisor. Although there is a superficial resemblance, this method is not the
bootstrap method (Efron and Gong 1983).
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C. L. KLIPP and L. MAHRT
Random
Original Data
1:
w θ 1
2:
w θ N:
w θ N
..
.
u∗1
2
∂U /∂z1
1:
w θ 6
u∗27
∂U /∂z14
w θ u∗61
..
.
∂U /∂z59
..
.
u∗35
∂U /∂z10
u∗2
..
.
∂U /∂z2
..
.
2:
u∗N
∂U /∂zN
N:
..
.
48
w θ 72
Figure 2. Example of construction of random dataset. See text for explanation.
TABLE 2.
S ELF - CORRELATION ANALYSIS
Level (m)
N
Rdata
Rrand
2 − R 2 Rdata
rand
0.5
1.5
5
10
20
30
40
50
45
34
88
82
77
69
92
86
0.75
0.73
0.83
0.85
0.71
0.95
0.92
0.89
0.61
0.71
0.65
0.67
0.66
0.67
0.70
0.67
0.17
0.02
0.27
0.28
0.06
0.45
0.34
0.35
Linear-correlation coefficient for the original data, Rdata , and
the average correlation coefficients for 1000 trials of random
data, Rrand , for each level. Rrand is the self-correlation and is
the adopted reference point of no correlation. Physical percent
variance explained by a linear model is approximated by the
2 − R 2 difference Rdata
rand . N is the number of data points.
For CASES99, the randomized data at different levels yield average values
for Rrand of 0.61–0.69 with standard deviations of 0.07–0.13 depending on level.
Although the values of Rdata are quite large (0.71–0.93), they are only modestly larger
than the reference point of no physical correlation, Rrand (Table 2).
(b) Variance explained
The square of the linear-correlation coefficient is a measure of the percent of the
variance in the data that is explained by a linear model (e.g. Snedecor and Cochran
1976). For the present datasets, most of the variance explained for the original data is
associated with self-correlation. The difference between the variance explained for the
2 , and the random data, R 2 original data, Rdata
rand , is proposed as a measure of the true
physical variance associated with the underlying processes governing the turbulence
(Table 2). This difference is not a true variance in that it can be negative if the original
2
data correlation, Rdata , and the self-correlation Rrand are of opposite sign and Rdata
is
2
less than R rand . Such a situation did not occur here.
This measure is not complete since the self-correlation is not necessarily expected
to be linear because φm ∝ 1/u∗ and z/L ∝ 1/u3∗ . In addition, the physical correlation
may not be linear for highly stable conditions. Although the linear correlation coefficient
can be sensitive to a few outlying points, we dismiss more sophisticated statistical
measures because of difficulties of interpretation and non-generality.
(c) Self-correlation: other datasets
Because the self-correlation coefficients are large for the CASES99 data, we
extended the study to other datasets with adequate estimates of vertical gradients to
determine if they yield similar results. Six stable nights in Borris95 provide about
FLUX–GRADIENT RELATIONSHIP IN THE SBL
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TABLE 3.
S ELF - CORRELATION ANALYSIS OF
B ORRIS 95 DATA
Level (m)
N
Rdata
Rrand
7
10
20
31
36
39
0.77
0.91
0.83
0.63
0.62
0.43
Linear-correlation coefficients Rdata for Borris95
data, and the average linear-correlation coefficients
for the random data, Rrand , for each level. N is
the number of data points.
TABLE 4.
S ELF - CORRELATION ANALYSIS OF
M ICROFRONTS DATA
Level (m)
N
Rdata
Rrand
3
10
107
101
0.92
0.88
0.65
0.67
See Table 3.
40 hours of data. This two-week field programme was conducted by the Meteorology
Department of the Risoe National Laboratory in July 1995 over a 4 km ×4 km area of
low heather in the Borris Moor of the central part of the Jutland Peninsula of Denmark.
Sonic anemometer data are analysed from three levels: 7 m (Solent Gill), 10 m (Kaijo
Denki B) and 20 m (Kaijo Denki B). Mean winds are from cup anemometers at 2, 10 and
20 m. There is insufficient neutral data to estimate z0 . Only data satisfying the conditions
outlined in section 4 are considered.
The values for Rdata for Borris95 are comparable to the CASES99 values.
The values of Rrand are also similar to the CASES99 values, except at 20 m, where it
is only 0.43 (Table 3). The data at 20 m sample a broader range of heat fluxes and shears
than sampled by the lower instruments, so the Rrand value is smaller even though the
range of u∗ values is similar at all the levels.
The Microfronts field campaign took place over grassland in south central Kansas,
late February to March 1995 (Sun 1999). The local-fit wind profiles use cup anemometer
data from 3, 5 and 10 m as well as a surface constraint of vanishing wind at z0 = 0.03 m.
Sonic anemometers were located at 3 and 10 m. The values for Rdata and Rrand are
comparable to the CASES99 analysis (Table 4).
6.
A DDITIONAL APPROACHES
(a) Self-correlation analysis of subsets of the CASES99 data
Since φm = 1 + βz/L is expected to be most valid at small z/L and may be less
accurate for large z/L, the data for each observational level are divided into two subsets:
one for z/L less than a cut-off value, the other for z/L greater than a cut-off value.
The cut-off value is chosen so that the two subsets have roughly the same number of
data points, so a different cut-off value is needed for each observational level (Table 5).
The test for self-correlation is repeated for each subset separately.
The Rdata values are larger for the data with small z/L (Table 5) indicating a better
fit to theory for the small z/L data than for the higher stability data. The 10 m level is
an exception for unknown reasons. The average random correlations Rrand are also
larger for the small z/L data and smaller for the more stable large z/L data. As a result,
2 − R 2 the physical variance, Rdata
rand , is about the same for both stability categories.
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C. L. KLIPP and L. MAHRT
TABLE 5.
S ELF - CORRELATION ANALYSIS FOR SUBSETS OF CASES99 DATA
Small z/L
Large z/L
Level (m)
Cut-off z/L
Nlow
Rdata
Rrand
Nhigh
Rdata
Rrand
0.5
1.5
5
10
20
30
40
50
0.08
0.12
0.50
0.90
1.70
1.30
1.84
1.41
21
19
39
44
36
31
50
46
0.84
0.95
0.96
0.74
0.78
0.94
0.90
0.91
0.72
0.75
0.83
0.80
0.72
0.77
0.75
0.66
24
15
49
38
41
38
42
40
0.49
0.57
0.62
0.79
0.62
0.91
0.89
0.80
0.58
0.72
0.56
0.63
0.61
0.61
0.64
0.54
Linear-correlation coefficient Rdata for small z/L data (see text) and large z/L data, and
the average linear-correlation coefficient for 1000 trials of random data, Rrand , for each
subset at each level. Nlow is the number of data points with z/L less than the cut-off z/L
value, Nhigh is the number with z/L more than the cut-off.
14
12
Im
10
8
6
4
2
0
0
2
4
6
8
10
z/L
Figure 3. 5 m data and randomized data. The line is φm = 1 + 4.7(z/L), white squares are the 88 points of data,
and the dark circles are 900 of the 1000 random points generated from the data. The 100 random data points not
shown have 10 < z/L < 60 and 1.5 < φm < 15. See text for explanation of symbols.
The small z/L data represent a wider range of u∗ values yielding a larger coefficient of
variation (section 5) than the large z/L data, resulting in a larger self-correlation Rrand
for the small z/L data.
(b) Large sample of random data
Apparently, the large fraction of total variance explained by self-correlation is
common not just to all three field campaigns, but also to subsets of the data. However,
even for the subset analysis, the Rrand values could be influenced by a small number
of very large z/L points generated by the randomization. Therefore, we analyse a single
large randomized dataset of 1000 points instead of the ensemble of small randomized
datasets (1000 sets of 88 points each). The same random sampling algorithm is used to
generate the 1000 random points, here from the 5 m observed data (Fig. 3). The large
number of random data points allows a visual sense of the influence of self-correlation.
In spite of substantial scatter in the random data (Fig. 3), the small z/L points
cluster nearer the theoretical relationship φm = 1 + 4.7(z/L) than the large z/L points.
FLUX–GRADIENT RELATIONSHIP IN THE SBL
2095
Although the random values of φm are never larger than 15, the random values of
z/L become as large as 60. The R value for all 1000 points is 0.65, the same as
the ensemble Rrand value for the same level (Table 2). Approximately 10% of the
random data are characterized by z/L > 10 while the 88 observed data points are all
z/L < 6. The ensemble Rrand values are influenced by these extremely large z/L
points. Even using only a weakly stable subset of data to generate the random data,
as in the previous subsection, will produce some random data points with large z/L
values.
Truncating the large random dataset to exclude z/L > 6 yields an R for the remain2 − R 2 ing 829 points of 0.71 resulting in a physical variance estimate of Rdata
rand =
0.18, even smaller than the original estimate of 0.27 in Table 2. Results are similar
repeating the ensemble Rrand analysis but restricting all 88 points in each of the 1000
realizations to z/L < 6:
Rrand = 0.69,
2
Rdata
− R 2 rand = 0.21.
Since self-correlation Rrand was increased by discarding the random data points
with extremely large z/L, the quantitative estimate of the self-correlation is subject to
some uncertainty. However, truncating the large random dataset or limiting z/L for the
ensemble analysis makes the analysis no longer truly random and should be interpreted
cautiously.
The 5 m data, plotted with the random data in Fig. 3, show excellent agreement
with theory in the range 0 < z/L < 0.5, and good agreement but increased scatter
in the range 0.5 < z/L < 1.0. As z/L increases further, the observed data become
nearly as scattered as the random points. At the other instrument levels, the scatter
of the observed data approaches the random points as z/L exceeds a threshold which
varies depending on the level. These subjective impressions are difficult to demonstrate
objectively (section 6(a)). Self-correlation is evident for all z/L based on the results of
Table 2, but is most significant for large z/L (Fig. 3) where the scatter for the original
data is as large as that for the randomized data. The decrease of φm below the linear
prediction found in strongly stable data could be due primarily to self-correlation.
7.
C ONDITIONAL ANALYSIS OF INTERMITTENT DATA
(a) Non-stationarity and intermittency
Turbulent scaling laws assume stationary conditions, so the non-stationarity due to
intermittent turbulence may introduce scatter. Removing subsections of weak turbulence
from each hour might reduce the scatter. The low-turbulence subsegments were too
contaminated by trend due to mesoscale motions to confidently analyse the data.
With very weak turbulence signals, even modest mesoscale trend becomes important.
Using the same hours of data as used for section 5 and Table 2, for each hour, a 100 s
section of data was considered turbulent if σw > 0.05 m s−1 . The 1-hour average flux is
taken as the average over all the turbulent sections of that hour. Each hour has at least
10 min (six 100 s segments) of data above the threshold. This removal of low-turbulence
sections from each hour reduces the non-stationarity due to intermittency. This method
differs from the technique in section 4 where entire hours were rejected as being below
threshold.
Conditional gradients are computed from locally fitted 100 s gradients based on
100 s values of U . The 1-hour gradient is taken as the average of only those 100 s
gradients from the turbulent sections of that hour. For the fully turbulent hours, this
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C. L. KLIPP and L. MAHRT
3
a)
Im
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.3
0.4
0.5
z/L
3
b)
Im
2
1
0
0
0.1
0.2
z/L
Figure 4. (a) Traditional and (b) conditional analysis of 0.5 m CASES99 data. The line is φm = 1 + 4.7(z/L),
open circles are intermittent data, filled circles are non-intermittent data that do not change with conditional
analysis. See text for meaning of symbols.
method produces the same gradient as the conventional method of using the 1-hour
mean wind to calculate the 1-hour gradient.
(b) Influence on scaling laws
The conditional analysis tends to significantly move the outlying points in the
relationship between φm and z/L towards the theoretical relationship as well as
move many of the other points a small distance towards the theoretical relationship.
These adjustments are mostly due to reduced z/L (compare panels (a) and (b) in Figs. 4
and 5). Both u2∗ and w θ increase by about the same percentage with application of conditional analysis, resulting in decreased z/L since L is proportional to u3∗ .
Conditional analysis increases Rdata most dramatically at 0.5 and 1.5 m (Table 6).
At these levels, conditional analysis increases the shear more than at the other levels. The
physical percent variance explained increases substantially at the lower levels owing to
conditional analysis, less so at higher levels.
2097
FLUX–GRADIENT RELATIONSHIP IN THE SBL
12
a)
10
Im
8
6
4
2
0
0
1
2
3
4
5
6
4
5
6
z/L
12
b)
10
Im
8
6
4
2
0
0
1
2
3
z/L
Figure 5.
TABLE 6.
As Fig. 4, but for 5 m CASES99 data.
S ELF - CORRELATION ANALYSIS OF CONDITIONAL DATA
Level (m)
N
Rdata
Rrand
2 − R 2 Rdata
rand
0.5
1.5
5
10
20
30
40
50
45
34
88
82
77
69
92
86
0.91
0.97
0.88
0.82
0.84
0.96
0.90
0.96
0.64
0.73
0.65
0.67
0.64
0.67
0.69
0.68
0.41
0.41
0.34
0.22
0.30
0.47
0.34
0.45
Linear-correlation coefficient Rdata for conditional data and the average
correlation coefficients for 1000 trials of random data, Rrand , for each
level. Rrand is the self-correlation and is the adopted reference point
of no correlation. Physical percent variance explained by a linear model
2 − R 2 is approximated by the difference Rdata
rand . N is the number of
data points.
2098
C. L. KLIPP and L. MAHRT
Conditional analysis most affects the data points with very stable, weak-wind
conditions. The CASES99 data, as well as the Borris95 and Microfronts data, have
many more hours with significant wind and weak stability than hours with weak winds.
Conditional analysis may have a larger effect for data dominated by weak-wind very
stable conditions with more extensive intermittency.
(c) Sensitivity to σw threshold value and σw averaging window
Sensitivity of the conditional analysis to the σw threshold value is now investigated.
Repeating the conditional analysis with different threshold values finds that the linearcorrelation coefficients for the conditional data change by only about 5% over a range
of 0.040 m s−1 < σw < 0.100 m s−1 at all the observational levels except the 0.5 m
level where R values vary by as much as 12% over the same range of σw . We are using
σw = 0.05 m s−1 as the threshold for this study.
Using short 100 s averaging windows to compute the conditional fluxes also allows
the conditional analysis to capture more detail of the intermittency compared to using
longer averaging windows. Repeating the analysis, using 1 min blocks of data to
compute σw and the conditional fluxes, yields results nearly identical to using 100 s
blocks. Windows shorter than about one minute do not capture enough turbulent flux
to be meaningful even for highly stable conditions. Using 5 min blocks results in more
scatter for both conditional and traditional data, the hours with small flux values being
most affected. The perturbations computed as deviations from longer averages are more
contaminated by mesoscale motions (Vickers and Mahrt 2003).
8.
z- LESS ANALOG TO M ONIN –O BUKHOV SIMILARITY THEORY
Self-correlation is difficult to avoid because most scaling schemes, in order to maintain simplicity, deliberately use a limited number of fundamental variables, requiring
the use of at least one of the fundamental variables in more than one of the resulting
dimensionless terms. This section evaluates the relative impact of self-correlation on
z-less similarity theory. To derive z-less similarity theory, one must assume that the
turbulence is stationary and not influenced by advection, the shear and wind vectors are
aligned, and that important additional length-scales are not introduced by curvature of
the temperature and wind profiles, as might occur with a low-level jet. Only conditional
data have been used in the following analysis as they yield less scatter compared to the
original data (section 7).
(a) Application of Buckingham Pi formalism
Similar to the derivation of Monin–Obukhov similarity theory in section 2, an
analogous pair of z-less Pi groups can be derived using
w θ ,
2
2
u∗ = (u w + v w )1/4 ,
∂U /∂z,
∂θ/∂z
and g/θ
as the relevant variables. Here, w θ and u∗ are the local fluxes at height z, not the
surface fluxes. These fundamental variables do not include z or the surface fluxes, so
the derivation is applicable to very stable conditions when the turbulence has become
partially decoupled from the surface and no longer directly related to the surface values
(Derbyshire 1999; Mahrt and Vickers 2003, and references therein). The constraints put
on this set of variables by Buckingham Pi theory will yield only two independent Pi
groups (e.g. Stull 1988; Barenblatt 1996).
FLUX–GRADIENT RELATIONSHIP IN THE SBL
2099
The terms, which comprise the two Pi groups, can be rearranged into a variety of
combinations. One familiar pair of Pi groups is
2
g ∂θ
∂U
≡ Ri,
(3)
θ ∂z
∂z
κL ∂U
≡ φm
,
u∗ ∂z
where L is explicitly calculated from local flux values as
(4)
−θ u3∗
.
(5)
κg w θ The first Pi group is the gradient Richardson number and the second is a dimension is also the inverse of the flux Richardson
less shear analogous to φm (Eq. (2)); φm
number. These equations are z-less in that no dependence on z or the boundary-layer
depth appear. This pair of Pi groups is not unique. Another possible pair is Ri and
φh ≡ (κL/θ∗ )(∂θ/∂z).
The formalism of Buckingham Pi theory states that if only two independent Pi
groups exist, one must be a function of the other as long as the variables chosen as
the relevant variables are the primary variables governing the flow. In other words, the
dimensionless shear must be a function of only the gradient Richardson number or vice
versa. Buckingham Pi theory does not give any indication of what that function might
be, so it must be determined experimentally.
.
In this case, K theory provides insight into the relationship between Ri and φm
Since φm is the inverse of the flux Richardson number and Ri is the gradient Richardson
in terms of the inverse
number, application of K theory yields an expression for φm
of Ri.
−Km (∂U /∂z)
Km −1
∂U
−u2∗
∂U
−1
φm = Rf =
=
=
Ri ,
(6)
∂z
∂z
Kh
−(g/θ )Kh (∂θ/∂z)
(g/θ )w θ
L=
where Km and Kh are the eddy diffusivity for momentum and heat, respectively, and Rf
is the flux of Richardson number.
The turbulent Prantl number Km /Kh is often taken to be 1.0 in the surface layer.
Most of our data support Km /Kh = 1.0, corresponding to a slope of unity in Fig. 6.
In general, the data would be plotted in the form of Rf vs. Ri, but this derivation
using Buckinham Pi formalism instructs us to also look at other ways to plot the data:
Rf −1 vs. Ri; Rf vs. Ri−1 ; Rf −1 vs. Ri−1 . On the traditional Rf vs. Ri plots, the data
support Km /Kh = 1.0. On the plots using Rf −1 vs. Ri−1 , the data that were clustered
near the origin on Rf vs. Ri plots (near-neutral data), are now spread out making it easier
to visualize the deviation of the near-neutral data from Km /Kh = 1.0. However, for the
near-neutral data, Monin–Obukhov similarity theory performs well, implying that z is
an important variable and that the above z-less analysis does not apply.
It is unknown why some of the data lie along the Km /Kh = 0.4 line. Although
conservative estimates of the measurement uncertainties do not support Km /Kh = 1
for these subsets, we can not rule out simultaneous overestimation of shear and
underestimation of the temperature gradient for this subset. Such errors would act to
systematically increase Ri−1 . However, for these weakly stable cases, Monin–Obukhov
similarity theory or local similarity may perform well, implying that z is an important
variable and that the above z-less analysis is incomplete.
2100
C. L. KLIPP and L. MAHRT
200
a)
Icm
150
100
50
0
0
50
100
Ri
30
150
200
-1
b)
25
Icm
20
15
10
5
0
0
5
10
15
Ri
20
25
30
-1
as a function of Ri−1 . The slope is K /K . The solid lines are K /K = 1, the dashed lines are
Figure 6. φm
m
h
m
h
KM /KH = 0.4. (a) Data for 0.5 m (3), 1.5 m () and 5 m (∗). (b) Data for 10 m (), 20 m (×), 30 m (), 40 m
(+), 50 m (2). See text for explanation of symbols.
(b) Self-correlation
Since ∂U /∂z occurs in both terms, this relationship is also subject to selfcorrelation. The average linear-correlation coefficients for the data range from 0.61–0.99
for the original data and 0.03–0.14 for the randomized data (Table 7), as compared
to 0.6–0.7 (Table 6) for Monin–Obukhov similarity theory or local similarity theory.
The small self-correlation is due to the fact that the range of shear data is relatively
small compared to the range of the turbulent fluxes (section 5). The largest range of shear
values is found at the lowest levels and is reflected in the larger Rrand values at 0.5 and
1.5 m. Monin–Obukhov similarity is much more vulnerable to self-correlation than the
z-less theory previously described due to the large range of the common variable u∗ .
FLUX–GRADIENT RELATIONSHIP IN THE SBL
TABLE 7.
2101
S ELF - CORRELATION ANALYSIS OF z- LESS SCALING
Level (m)
N
Rdata
Rrand
2 − R 2 Rdata
rand
0.5
1.5
5
10
20
30
40
50
45
28
80
75
67
59
82
76
0.93
0.998
0.985
0.94
0.63
0.83
0.61
0.88
0.14
0.10
0.03
0.04
0.05
0.06
0.05
0.07
0.85
0.986
0.97
0.88
0.39
0.69
0.37
0.77
as a function of
Linear-correlation coefficient Rdata for shear φm
Ri−1 and the average correlation coefficients for 1000 trials of random data, Rrand and the estimate of the percent physical variance
2 − R 2 explained, Rdata
rand . Rrand is the amount of correlation
due to self-correlation and is the reference point of no correlation
rather than R = 0.
9.
C ONCLUSIONS
For stable conditions, self-correlation contributes significantly to the linearcorrelation coefficient between the non-dimensional shear φm and z/L owing to the
occurrence of u∗ in both terms. The self-correlation, as measured by the average
linear-correlation coefficient for randomized data, was found to be significantly larger
than zero and typically not small compared to the linear correlation for the original
data. Self-correlation was large for the entire range of stability and was large whether
Monin–Obukhov similarity was valid or whether only local similarity theory was valid.
Subjective examination of actual data compared to a large sample of randomized data
seems to imply an excellent fit of the near-neutral data to the linear theory, but an unexpectedly large self-correlation for near-neutral conditions made this impression difficult
to support objectively. Although the estimate of self-correlation is itself an uncertain
process, we conclude that substantial physical correlation can not be established for our
three stable boundary-layer datasets.
While the physical correlation (excess correlation beyond self-correlation) between
φm and z/L is generally much smaller than the total correlation, linear-correlation
analysis may be incomplete. There is a need to develop a better statistical approach
that will objectively quantify the effects of self-correlation. Furthermore, the correlation
may be reduced by observational errors especially for very stable conditions. The selfcorrelation for z-less similarity theory based on the Richardson number was much less
than that for Monin–Obukhov similarity theory, although still significant near the ground
surface.
Removal of record subsections corresponding to very weak turbulence (conditional
analysis) improves the flux–gradient relationship and reduces non-stationarity such
that the conditions required for similarity are likely to be met. With such restrictions,
similarity theory explains significantly more of the variance of φm , depending on
observational level. Such conditional analysis may show more dramatic reductions in
scatter for data more dominated by intermittent turbulent conditions (weaker winds).
ACKNOWLEDGEMENTS
We gratefully acknowledge the field assistance of the National Centers for Atmospheric Research ATD staff, Jielun Sun and Sean Burns in CASES99 and the comments of two anonymous reviewers. Hans Jørgensen provided the data from Borris95.
This material is based upon work supported by Grant DAAD19-0210224 from the Army
2102
C. L. KLIPP and L. MAHRT
Research Office and Grant 0107617-ATM from the Physical Meteorology Program of
the National Sciences Program. Thanks to Dean Vickers of Oregon State University for
the roughness-length calculations.
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